and it was therefore necessary to recalculate them a t least approximately to isothermal data. Assuming that the dependence of the ratio of activity coefficients on the composition does not vary much with temperature, we can calculate from this dependence the relative volatilities and consequently also the composition of the vapor phase a t the desired temperature. The dependence of the ratio of activity coefficients on the composition was correlated on a computer by using statistical weights (Wichterle, 1966). The experimental and calculated results are plotted as an x-y relationship in Figure 4.
the dependence of the height of the integral curve on the mass of the vapor sample. The results obtained were utilized for determination of vapor-liquid equilibrium for the system hexane-toluene. The equilibrium data were correlated by a third-order Margules equation, the mean deviation in the composition of the vapor phase being 1.3 mole %. literature Cited
Dreisbach, R. R., Aduan. Chem. Ser., No. 15, 11 (1955). Dreisbach, R. R., Advan. Chem. Ser., No. 22, 19 (1959). Grubner, O., Czech. Patent 121,672 (1967). Hbla, E., Pick, J., Fried, V., Villm, O., “Vapour-Liquid Equilibrium,” Pergamon Press, Oxford, 1967. Sieg, L., Chem.-Zng. Tech. 22, 322 (1950). Wichterle, I., Collection Czech. Chem. Commun. 31, 3821 (1966). 2, 155 Wichterle, I., Hbla, E., IND.ENC.CHEM.FUNDAMENTALS (1963).
Discussion
The semimicromethod for determining the variables which characterize vapor-liquid equilibrium is rapid and in combination with gas chromatographic analysis it gives good results. By standardizing the conditions for sampling the vapor phase, the analyses were made reproducible to 2.8%. In the present work we measured over the range of interest
RECEIVED for review January 18, 1968 ACCEPTEDNovember 20, 1968
COMMUNICAT IONS PREDICTION OF POLYSTYRENE MELT TENSILE BEHAVIOR
-
A reasonably accurate prediction of tensile viscosity from dynamic and simple shearing viscosities is made for a polystyrene melt using a llnetwork rupture’’ theory of rheological behavior.
RELATIVELYfew measurements have been made of the
extensional (or Trouton) viscosity, p t , of polymer melts and solutions, but it has been found (Ballman, 1965) that pt is often much greater than ps, the shear viscosity, at equal strain rates. The extensional viscosity is defined for a circular rodlike specimen as the tensile stress divided by the rate of extension GI = au/az, where z is measured along the rod center line. The velocity field is assumed to be v = (Gz, -Gy/2, -Gz/2), the material being incompressible. It is readily shown that for a Newtonian fluid (Trouton, 1906) 3P8 (1) Figure 1 (circles) shows experimental measurements of pt and pe for a polystyrene melt described previously (Ballman, 1965). The simple shearing rate (du/dy) is denoted by y. The high value of pt/pe when compared at y = G is obvious and here we attempt to predict p t ( G ) given p 8 ( y ) and the dynamic small-strain viscosity, Pd ( w ) . The dynamic viscosity, Pd, is a function of the frequency, w (rad./sec.), of the applied small sinusoidal shear. Suppose that the velocity vector v = (ydeht, 0, 0 ) in a small strain motion, and that the z - y shear stress component is written ?eiot where i is a complex number. Pd is defined as the real part of the complex viscosity, p* ( w ) , where Pt =
.3 = p*ci (2 1 For a Newtonian fluid ps = Pd = p*. Pd for the polystyrene sample was measured on a Weissenberg rheogoniometer and is shown in Figure 1 (circles). To predict p t we use a “network-rupture” type of constitutive relation (Tanner and Simmons, 1967) which is a simple form of BKZ fluid. This gives the well-known expression for Fd in terms of n discrete relaxation times, A,:
an(1
Pd(w) = n
588
I&EC
+
FUNDAMENTALS
(3 1
10’
ioe
-
\
.-0
a 10’
:10‘ 0 0
> IO5
-
’X
EXPT.
-X- FITTED CURVES --- PREDICTED, B = 1 2
FREQUENCY OR
STRAIN
\
kw
RATE
\
w(rad/sec.) y,G (seCi)
Figure 1. Measured and predicted properties of polystyrene melt
where a, are constants with the dimensions of viscosity. Choosing X,+1= A,/ an iterative numerical procedure with smoothing (Tanner, 1968) was used to assign the a,. The longest relaxation time was taken to be 505 seconds and 13 terms were used in Equation 3. The crosses in Figure 1 are points recalculated with the fitting values of G. Choosing a value B of the rupture strain parameter the shear viscosity,
m,
may be found from the result (Tanner and Simmons, 1967)
pa,
cc
(
1 - C anC1 -
S T -
n
(1
+
~ X P(-B/Xn./)I
(4)
The curve for B = 12 is fairly satisfactory (Figure 1). Although the errors might be reduced by letting B vary with n, this is not attempted here because the lack of information a t very low shear rates and frequencies leads to some uncertainties. From the knowledge of A,, an, and B already given, the elongational viscosity may be predicted under certain assumptions concerning the rupture of the network. The simplest assumption (Tanner and Simmons, 1967) gives the result p =
C aflrn-l
from a t least one point of view (Tanner, 1966), the tensile and shearing flows are the most divergent steady homogeneous flows possible in an incompressible medium. Nomenclature
6 a,
-
B, B = rupture strain magnitudes G = extensional rate, d u / d z , see.-’ =
n
= integer
q
= integer
V
r,
+
+
This may be solved numerically for a given value of B. Calculation using B = 12 and Equations 5 and 6 gives the predicted Trouton viscosity (Figure 1). Depending on the purpose of the exercise, the prediction may or may not be considered satisfactory. If one considers the limited amount of experimental information usually available in practice and the need to make reasonable predictions of moderate accuracy, the rupture theory seems to be helpful for engineering purposes. The theory can be expected to work equally well for any steady homogeneous flow, since
E
t, t’ = times, sec. u, v, w = components of velocity vector, cm./sec. y
where r = AnG. To relate the value of B to B it is assumed that the trace of the Finger strain measure (Lodge, 1964) governs rupture of the network. The strain matrix is computed relative to the present time, t, as reference; a t a time t’ in the past it may readily be shown (Lodge, 1964) that in simple shearing the trace of the strain matrix is y 2 ( t- t ’ ) ; when this quantity reaches a value of BZ,rupture occurs. Similarly, in elongational flow the quantity G(t - t ’ ) occurs in the calculation. The trace of the strain matrix may be computed and equated to B2; one finds the equation for the rupture parameter B, which defines the rupture time, t‘, as 3 B2 = exp 2 B 2 exp - B Hence, (6 ) B-lnBforB>>l
47
i
5 , y,
n
= amplitude of sinusoidal shear rate, sec.-l = strength of nth time constant, poises
pd ps pt p* Xn
i;
w
= velocity vector, cm./sec. z = coordinates, cm. = shear rate, see.-’ = = X,G = dynamic viscosity, poises = shearing viscosity, poises = tensile viscosity, poises = complex viscosity, poises = nth time constant, sec. = complex amplitude of shear stress, dynes/sq. cm. = frequency, rad./sec.
literature Cited
Ballman, R. L., Rheol. Acta 4, 137 (1965). Lodge, A. S., “Elastic Liquids,” Academic Press, New York, 1964.
Tanner, R. I., IND.E m . CHEM.F U X D A M E N T6,A55 L ~(1966). Tanner, R. I., J. A p p l . Polymer Sei. 12, 1649 (1968). Tanner, R. I., Simmons, J. M., Chem. Eng. Sci. 22, 1803 (1967). Trouton, F. T., Proc. Roy. SOC.A77, 426 (1906). ROGER I. TANNER Brown University Providence, R.I. 0.9912 RICHARD L. BALLMAN Momanto Co. Springjield, Mass. RECEIVED for review February 15, 1968 ACCEPTEDMarch 3, 1969 Investigation supported in part by the National Aeronautics and Space Administration under the hlultidisciplinary SpaceRelated Research Program (Grant NGR40-002-009) at Brown University.
CORRELATION OF DIFFUSION COEFFICIENTS FOR PARAFFIN, AROMATIC, AND CYCLOPARAFFIN HYDROCARBONS IN WATER Diffusion coefficients for all hydrocarbons investigated can be correlated over a temperature range of 2’ to using the Wilke-Chang empirical equation.
60’ C.
U N T I L recently, very few data on the diffusion of hydrocarbons in water have been available. Kartsev et al. (1959) reported work credited to Antonov for the diffusion of methane, ethane, propane, and *hexane in water. Recently, diffusion coefficients were reported for methane at 25’, 45’, and 65’ C. (Gubbins et al., 1966). With regard to other hydrocarbons, diffusion data have been reported
only for acetylene in water (Tammann and Jensen, 1929), propylene in water (Vivian and King, 1964), and ethylene, propylene, and butylene in water (Unver and Himmelbau, 1964). A project has therefore been under way for some time in this laboratory to study the diffusion through water of the various hydrocarbons found in petroleum. Results have been VOL.
8
NO.
3
AUGUST
1969
569
